Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 073, 16 pages      arXiv:1904.09323

A Kähler Compatible Moyal Deformation of the First Heavenly Equation

Marco Maceda and Daniel Martínez-Carbajal
Departamento de Física, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, C.P. 03340, Deleg. Iztapalapa, Mexico City, México

Received June 07, 2019, in final form September 08, 2019; Published online September 22, 2019

We construct a noncommutative Kähler manifold based on a non-linear perturbations of Moyal integrable deformations of $D=4$ self-dual gravity. The deformed Kähler manifold preserves all the properties of the commutative one, and we obtain the associated noncommutative Kähler potential using the Moyal deformed gravity approach. We apply this construction to the Atiyah-Hitchin metric and its Kähler potential, which is useful in the description of interactions among magnetic monopoles at low energies.

Key words: heavenly equations; Moyal deformation; Atiyah-Hitchin metric.

pdf (368 kb)   tex (26 kb)  


  1. Alexandrov S., Pioline B., Vandoren S., Self-dual Einstein spaces, heavenly metrics, and twistors, J. Math. Phys. 51 (2010), 073510, 31 pages, arXiv:0912.3406.
  2. Alvarez-Gaumé L., Freedman D.Z., Geometrical structure and ultraviolet finiteness in the supersymmetric $\sigma $-model, Comm. Math. Phys. 80 (1981), 443-451.
  3. Atiyah M.F., Hitchin N.J., Low energy scattering of nonabelian monopoles, Phys. Lett. A 107 (1985), 21-25.
  4. Atiyah M.F., Hitchin N.J., The geometry and dynamics of magnetic monopoles, M.B. Porter Lectures, Princeton University Press, Princeton, NJ, 1988.
  5. Atiyah M.F., Hitchin N.J., Singer I.M., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461.
  6. Bakas I., Remarks on the Atiyah-Hitchin metric, 3.0.CO;2-7">Fortschr. Phys. 48 (2000), 9-14, arXiv:hep-th/9903256.
  7. Bakas I., Sfetsos K., Toda fields of ${\rm SO}(3)$ hyper-Kähler metrics and free field realizations, Internat. J. Modern Phys. A 12 (1997), 2585-2611, arXiv:hep-th/9604003.
  8. Boyer C.P., The geometry of complex self-dual Einstein spaces, in Nonlinear Phenomena (Oaxtepec, 1982), Lecture Notes in Phys., Vol. 189, Springer, Berlin, 1983, 25-46.
  9. Boyer C.P., Finley III J.D., Killing vectors in self-dual, Euclidean Einstein spaces, J. Math. Phys. 23 (1982), 1126-1130.
  10. Boyer C.P., Plebański J.F., An infinite hierarchy of conservation laws and nonlinear superposition principles for self-dual Einstein spaces, J. Math. Phys. 26 (1985), 229-234.
  11. Douglas M.R., Nekrasov N.A., Noncommutative field theory, Rev. Modern Phys. 73 (2001), 977-1029, arXiv:hep-th/0106048.
  12. Eguchi T., Gilkey P.B., Hanson A.J., Gravitation, gauge theories and differential geometry, Phys. Rep. 66 (1980), 213-393.
  13. Esposito G., From spinor geometry to complex general relativity, Int. J. Geom. Methods Mod. Phys. 2 (2005), 675-731, arXiv:hep-th/0504089.
  14. Finley III J.D., Plebański J.F., Further heavenly metrics and their symmetries, J. Math. Phys. 17 (1976), 585-596.
  15. García D.A., Plebański J.F., Seven parametric type-$D$ solutions of Einstein-Maxwell equations in the basic left-degenerate representation, Nuovo Cimento A 40 (1977), 224-234.
  16. Gegenberg J.D., Das A., Stationary Riemannian space-times with self-dual curvature, Gen. Relativity Gravitation 16 (1984), 817-829.
  17. Gibbons G.W., Hawking S.W., Classification of gravitational instanton symmetries, Comm. Math. Phys. 66 (1979), 291-310.
  18. Gibbons G.W., Manton N.S., Classical and quantum dynamics of BPS monopoles, Nuclear Phys. B 274 (1986), 183-224.
  19. Gibbons G.W., Olivier D., Ruback P.J., Valent G., Multicentre metrics and harmonic superspace, Nuclear Phys. B 296 (1988), 679-696.
  20. Gibbons G.W., Ruback P.J., The hidden symmetries of multi-centre metrics, Comm. Math. Phys. 115 (1988), 267-300.
  21. Hanany A., Pioline B., (Anti-)instantons and the Atiyah-Hitchin manifold, J. High Energy Phys. 2000 (2000), no. 7, 001, 23 pages, arXiv:hep-th/0005160.
  22. Hitchin N.J., Karlhede A., Lindström U., Roček M., Hyperkähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535-589.
  23. Husain V., Self-dual gravity as a two-dimensional theory and conservation laws, Classical Quantum Gravity 11 (1994), 927-937, arXiv:gr-qc/9310003.
  24. Ionas R.A., Elliptic constructions of hyperkähler metrics I: The Atiyah-Hitchin manifold, arXiv:0712.3598.
  25. Ko M., Ludvigsen M., Newman E.T., Tod K.P., The theory of ${\mathcal H}$-space, Phys. Rep. 71 (1981), 51-139.
  26. Moyal J.E., Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 (1949), 99-124.
  27. Newman E.T., Porter J.R., Tod K.P., Twistor surfaces and right-flat spaces, Gen. Relativity Gravitation 9 (1978), 1129-1142.
  28. Olivier D., Complex coordinates and Kähler potential for the Atiyah-Hitchin metric, Gen. Relativity Gravitation 23 (1991), 1349-1362.
  29. Park Q.H., Self-dual gravity as a large-$N$ limit of the 2D non-linear sigma model, Phys. Lett. B 238 (1990), 287-290.
  30. Park Q.H., 2D sigma model approach to 4D instantons, Internat. J. Modern Phys. A 7 (1992), 1415-1447.
  31. Penrose R., Nonlinear gravitons and curved twistor theory, Gen. Relativity Gravitation 7 (1976), 31-52.
  32. Penrose R., The nonlinear graviton, Gen. Relativity Gravitation 7 (1976), 171-176.
  33. Penrose R., Ward R.S., Twistors for flat and curved space-time, in General Relativity and Gravitation, Vol. 2, Plenum, New York - London, 1980, 283-328.
  34. Plebański J.F., Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975), 2395-2402.
  35. Plebański J.F., Przanowski M., Rajca B., Tosiek J., The Moyal deformation of the second heavenly equation, Acta Phys. Polon. B 26 (1995), 889-902.
  36. Plebański J.F., Robinson I., Left-degenerate vacuum metrics, Phys. Rev. Lett. 37 (1976), 493-495.
  37. Plebański J.F., Torres del Castillo G.F., ${\mathcal{HH}}$ spaces with an algebraically degenerate right side, J. Math. Phys. 23 (1982), 1349-1352.
  38. Strachan I.A.B., The Moyal algebra and integrable deformations of the self-dual Einstein equations, Phys. Lett. B 283 (1992), 63-66.
  39. Strachan I.A.B., Hierarchy of conservation laws for self-dual gravity, Classical Quantum Gravity 10 (1993), 1417-1423.
  40. Strachan I.A.B., The symmetry structure of the anti-self-dual Einstein hierarchy, J. Math. Phys. 36 (1995), 3566-3573, arXiv:hep-th/9410047.
  41. Strachan I.A.B., A geometry for multidimensional integrable systems, J. Geom. Phys. 21 (1997), 255-278, arXiv:hep-th/9604142.
  42. Szabo R.J., Quantum field theory on noncommutative spaces, Phys. Rep. 378 (2003), 207-299, arXiv:hep-th/0109162.
  43. Takasaki K., Dressing operator approach to Moyal algebraic deformation of selfdual gravity, J. Geom. Phys. 14 (1994), 111-120, arXiv:hep-th/9212103.

Previous article  Next article  Contents of Volume 15 (2019)